Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(112,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([5, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.112");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 605.m (of order \(20\), degree \(8\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.83094932229\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
112.1 | −1.23116 | − | 2.41628i | −0.183968 | + | 1.16153i | −3.14709 | + | 4.33160i | −0.956130 | + | 2.02134i | 3.03306 | − | 0.985502i | −0.535369 | + | 0.0847942i | 8.98397 | + | 1.42292i | 1.53787 | + | 0.499685i | 6.06127 | − | 0.178306i |
112.2 | −1.05904 | − | 2.07848i | −0.318551 | + | 2.01125i | −2.02296 | + | 2.78437i | 2.18295 | + | 0.484484i | 4.51772 | − | 1.46790i | 4.54674 | − | 0.720133i | 3.32162 | + | 0.526094i | −1.09050 | − | 0.354324i | −1.30484 | − | 5.05032i |
112.3 | −1.05076 | − | 2.06223i | 0.159862 | − | 1.00933i | −1.97313 | + | 2.71579i | −1.98733 | + | 1.02495i | −2.24944 | + | 0.730888i | 2.99845 | − | 0.474908i | 3.10187 | + | 0.491288i | 1.85999 | + | 0.604346i | 4.20189 | + | 3.02136i |
112.4 | −0.890440 | − | 1.74759i | 0.459138 | − | 2.89888i | −1.08561 | + | 1.49421i | 0.260122 | − | 2.22089i | −5.47488 | + | 1.77890i | −2.07637 | + | 0.328864i | −0.296505 | − | 0.0469618i | −5.33954 | − | 1.73492i | −4.11281 | + | 1.52298i |
112.5 | −0.832753 | − | 1.63437i | −0.488992 | + | 3.08737i | −0.802115 | + | 1.10402i | −1.82276 | + | 1.29520i | 5.45312 | − | 1.77183i | 0.491081 | − | 0.0777795i | −1.15109 | − | 0.182315i | −6.43959 | − | 2.09235i | 3.63475 | + | 1.90048i |
112.6 | −0.679174 | − | 1.33295i | −0.340390 | + | 2.14914i | −0.139917 | + | 0.192579i | 0.485485 | − | 2.18273i | 3.09589 | − | 1.00591i | −2.32674 | + | 0.368519i | −2.60345 | − | 0.412346i | −1.64977 | − | 0.536043i | −3.23920 | + | 0.835322i |
112.7 | −0.544755 | − | 1.06914i | 0.0572260 | − | 0.361310i | 0.329265 | − | 0.453195i | 2.07465 | − | 0.834158i | −0.417466 | + | 0.135643i | 1.48586 | − | 0.235337i | −3.03420 | − | 0.480570i | 2.72590 | + | 0.885698i | −2.02201 | − | 1.76368i |
112.8 | −0.323023 | − | 0.633967i | 0.440005 | − | 2.77808i | 0.877999 | − | 1.20846i | −2.10640 | − | 0.750378i | −1.90335 | + | 0.618434i | 3.39829 | − | 0.538236i | −2.45526 | − | 0.388874i | −4.67097 | − | 1.51769i | 0.204700 | + | 1.57778i |
112.9 | −0.280420 | − | 0.550355i | −0.235077 | + | 1.48421i | 0.951315 | − | 1.30937i | −2.19624 | − | 0.420141i | 0.882765 | − | 0.286828i | −1.51988 | + | 0.240726i | −2.20753 | − | 0.349639i | 0.705537 | + | 0.229243i | 0.384643 | + | 1.32653i |
112.10 | −0.120168 | − | 0.235842i | 0.00828396 | − | 0.0523028i | 1.13439 | − | 1.56135i | 1.85452 | + | 1.24930i | −0.0133307 | + | 0.00433140i | −3.21766 | + | 0.509627i | −1.02742 | − | 0.162727i | 2.85050 | + | 0.926184i | 0.0717827 | − | 0.587500i |
112.11 | 0.120168 | + | 0.235842i | 0.00828396 | − | 0.0523028i | 1.13439 | − | 1.56135i | 1.85452 | + | 1.24930i | 0.0133307 | − | 0.00433140i | 3.21766 | − | 0.509627i | 1.02742 | + | 0.162727i | 2.85050 | + | 0.926184i | −0.0717827 | + | 0.587500i |
112.12 | 0.280420 | + | 0.550355i | −0.235077 | + | 1.48421i | 0.951315 | − | 1.30937i | −2.19624 | − | 0.420141i | −0.882765 | + | 0.286828i | 1.51988 | − | 0.240726i | 2.20753 | + | 0.349639i | 0.705537 | + | 0.229243i | −0.384643 | − | 1.32653i |
112.13 | 0.323023 | + | 0.633967i | 0.440005 | − | 2.77808i | 0.877999 | − | 1.20846i | −2.10640 | − | 0.750378i | 1.90335 | − | 0.618434i | −3.39829 | + | 0.538236i | 2.45526 | + | 0.388874i | −4.67097 | − | 1.51769i | −0.204700 | − | 1.57778i |
112.14 | 0.544755 | + | 1.06914i | 0.0572260 | − | 0.361310i | 0.329265 | − | 0.453195i | 2.07465 | − | 0.834158i | 0.417466 | − | 0.135643i | −1.48586 | + | 0.235337i | 3.03420 | + | 0.480570i | 2.72590 | + | 0.885698i | 2.02201 | + | 1.76368i |
112.15 | 0.679174 | + | 1.33295i | −0.340390 | + | 2.14914i | −0.139917 | + | 0.192579i | 0.485485 | − | 2.18273i | −3.09589 | + | 1.00591i | 2.32674 | − | 0.368519i | 2.60345 | + | 0.412346i | −1.64977 | − | 0.536043i | 3.23920 | − | 0.835322i |
112.16 | 0.832753 | + | 1.63437i | −0.488992 | + | 3.08737i | −0.802115 | + | 1.10402i | −1.82276 | + | 1.29520i | −5.45312 | + | 1.77183i | −0.491081 | + | 0.0777795i | 1.15109 | + | 0.182315i | −6.43959 | − | 2.09235i | −3.63475 | − | 1.90048i |
112.17 | 0.890440 | + | 1.74759i | 0.459138 | − | 2.89888i | −1.08561 | + | 1.49421i | 0.260122 | − | 2.22089i | 5.47488 | − | 1.77890i | 2.07637 | − | 0.328864i | 0.296505 | + | 0.0469618i | −5.33954 | − | 1.73492i | 4.11281 | − | 1.52298i |
112.18 | 1.05076 | + | 2.06223i | 0.159862 | − | 1.00933i | −1.97313 | + | 2.71579i | −1.98733 | + | 1.02495i | 2.24944 | − | 0.730888i | −2.99845 | + | 0.474908i | −3.10187 | − | 0.491288i | 1.85999 | + | 0.604346i | −4.20189 | − | 3.02136i |
112.19 | 1.05904 | + | 2.07848i | −0.318551 | + | 2.01125i | −2.02296 | + | 2.78437i | 2.18295 | + | 0.484484i | −4.51772 | + | 1.46790i | −4.54674 | + | 0.720133i | −3.32162 | − | 0.526094i | −1.09050 | − | 0.354324i | 1.30484 | + | 5.05032i |
112.20 | 1.23116 | + | 2.41628i | −0.183968 | + | 1.16153i | −3.14709 | + | 4.33160i | −0.956130 | + | 2.02134i | −3.03306 | + | 0.985502i | 0.535369 | − | 0.0847942i | −8.98397 | − | 1.42292i | 1.53787 | + | 0.499685i | −6.06127 | + | 0.178306i |
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
55.e | even | 4 | 1 | inner |
55.k | odd | 20 | 3 | inner |
55.l | even | 20 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 605.2.m.g | 160 | |
5.c | odd | 4 | 1 | inner | 605.2.m.g | 160 | |
11.b | odd | 2 | 1 | inner | 605.2.m.g | 160 | |
11.c | even | 5 | 1 | 605.2.e.c | ✓ | 40 | |
11.c | even | 5 | 3 | inner | 605.2.m.g | 160 | |
11.d | odd | 10 | 1 | 605.2.e.c | ✓ | 40 | |
11.d | odd | 10 | 3 | inner | 605.2.m.g | 160 | |
55.e | even | 4 | 1 | inner | 605.2.m.g | 160 | |
55.k | odd | 20 | 1 | 605.2.e.c | ✓ | 40 | |
55.k | odd | 20 | 3 | inner | 605.2.m.g | 160 | |
55.l | even | 20 | 1 | 605.2.e.c | ✓ | 40 | |
55.l | even | 20 | 3 | inner | 605.2.m.g | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
605.2.e.c | ✓ | 40 | 11.c | even | 5 | 1 | |
605.2.e.c | ✓ | 40 | 11.d | odd | 10 | 1 | |
605.2.e.c | ✓ | 40 | 55.k | odd | 20 | 1 | |
605.2.e.c | ✓ | 40 | 55.l | even | 20 | 1 | |
605.2.m.g | 160 | 1.a | even | 1 | 1 | trivial | |
605.2.m.g | 160 | 5.c | odd | 4 | 1 | inner | |
605.2.m.g | 160 | 11.b | odd | 2 | 1 | inner | |
605.2.m.g | 160 | 11.c | even | 5 | 3 | inner | |
605.2.m.g | 160 | 11.d | odd | 10 | 3 | inner | |
605.2.m.g | 160 | 55.e | even | 4 | 1 | inner | |
605.2.m.g | 160 | 55.k | odd | 20 | 3 | inner | |
605.2.m.g | 160 | 55.l | even | 20 | 3 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 146 T_{2}^{156} + 13159 T_{2}^{152} - 954708 T_{2}^{148} + 61579675 T_{2}^{144} + \cdots + 45\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).